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Simplified Method for Indoor Airflow Simulation Proceedings of CLIMA 2000 World Congress, Brussels, Belgium. Simplified Method for Indoor Airflow Simulation Qingyan Chen and Weiran Xu Building Technology Program, Department of Architecture Massachusetts Institute of Technology Room 5-418, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA Phone: 617-253-7714, Fax: 617-253-6152 E_mail: [email protected] URL: http://web.mit.edu/qchen/www/ ABSTRACT At present, numerical simulation of room airflows is mainly conducted by either the Computational-Fluid-Dynamics (CFD) method or various zonal/network models. The CFD approach needs a large capacity of computer and a skillful expert. The results obtained with zonal/network models have great uncertainties. This paper proposes a new simplified method to simulate three-dimensional distributions of air velocity, temperature, and contaminant concentrations in rooms. The method assumes turbulent viscosity to be a function of length-scale and local mean velocity. The new model has been used to predict natural convection, forced convection, mixed convection, and displacement ventilation in a room. The results agree reasonably with experimental data and the CFD computations. The simplified method uses much less computer memory and the computing speed is at least 10 times faster, compared with the CFD method. The grid number can often be reduced so that the computing time needed for a three-dimensional case can be a few minutes in a PC. INTRODUCTION Proper design of indoor environment requires detailed information of indoor air distribution, such as airflow pattern, velocity, temperature, and contaminant concentrations. The information can be obtained by experimental measurements and computational simulations. Experimental measurements are reliable but need large labor- effort and time. Therefore, the experimental approach is not feasible as a general design tool. Two approaches of computational simulations are available for the study of indoor air distribution. The first approach is the computational-fluid-dynamics (CFD) methods and the second is simplified flow simulation methods. Computational-Fluid-Dynamics Methods The CFD methods solve the Navier-Stokes equations for flows. For laminar flows 1 the computed results are accurate and reliable. However, it is difficult to predict turbulent flows. Very fine numerical resolution is required to capture all the details of the indoor turbulent flow. This type of simulation is direct numerical simulation. The direct numerical simulation for a practical flow needs a huge computer system that is not available. Indoor airflow simulations use turbulence models to compute the mean values. This can be done with the capacity and speed of present computers. Eddy-viscosity models are the most popular turbulence models. The CFD program with eddy-viscosity models solves air velocities, temperature, contaminant concentration, and turbulent quantities in a space. The space is divided into 10,000 to one million cells to achieve a reasonable accuracy for a three-dimensional flow problem. In addition, the CFD program user should have good knowledge of fluid dynamics, numerical technique, and indoor air distribution. However, a large computer and a skillful user will not guarantee success. Chen (1997) reported many failures in using the CFD method in a group with more than ten-year experience. Obviously, most HVAC designers and architects do not have the computer capacity and the CFD knowledge. Therefore, in predicting indoor air distribution and designing a comfortable indoor environment, application of the CFD method is limited. Simplified Flow Simulation Methods The second approach does not use a turbulence model. The approach uses a much coarse cell system. In most cases, the total cell number for a space is less than 10,000. A very simple method (Lebrun and Ngendakumana 1987) is to fix airflow patterns and use empirical flow laws for different flow components, such as jets, plumes, etc. In many cases, the airflow patterns are difficult to impose even by an experienced fluid dynamics engineer. The method has limited applications. Another popular method is the network model (Walton 1989). The model determines flow within a space by Bernoulli’s equation. The method works reasonably for parabolic flows and is useful to analyze combined problems of HVAC systems, infiltration, and multi-room airflow simultaneously. However, the uncertainty is large if the method is applied for a room presented by several different cells or sub-volumes. The method proposed by Wurtz and Nataf (1994) is to calculate indoor air pressure using a degraded equation for the momentum. The airflow between two zones is determined by the pressure differential. Because of the poor representation of the momentum, the method does not work for pressure and buoyancy driven flows, i.e. flows set up by temperature differences in the air. A recent zonal model developed by Inard et al. (1996) calculates flow rate for zones with small momentum through pressure distribution. Although the results are 2 consistent with experimental data, the model may not be applied for high momentum flows. In addition, the method uses a discharge coefficient that must be determined through experiment. When a room is subdivided by a partition wall or a large opening, all the above models use a discharge coefficient to calculate flow due to pressure or temperature difference. This will further reduce the reliability of the methods since a general form for the discharge coefficient has not been established. Calculations for new geometries require an CFD run or experiment to determine the discharge coefficient. Justification of Need Many HVAC design engineers and architects have limited knowledge of fluid flow and do not have the access to a large computer. It is important to develop a simplified model to simulate indoor airflow in a personal computer. The flow program should then be coupled with an energy analysis program to simulate simultaneously airflow, thermal comfort, and energy consumption of HVAC systems. The program will also allow the temperature of interior walls to be predicted. The program would serve as a tool to accurately provide design information and to properly size HVAC systems and assure comfort conditions exist at all important locations within the space. The goal of the present investigation is to develop a program which will provide design information to establish acceptable comfort conditions through the interior space. Precise rigor and exact predictions will be relaxed to allow the program to be easily used by HVAC engineers with a minimum of training and modest desktop personal computers. The following section describes a new simplified method. NEW SIMPLIFIED METHOD Governing Flow Equations Most indoor airflows are turbulent. Often airflow calculations use the Buossinesq approximation. The approximation takes air density as constant in the momentum terms and considers the buoyancy influence on air movement by the difference between the local air weight and the pressure gradient. With an eddy-viscosity model, the indoor airflows can be described by the following time-averaged Navier-Stokes equations for the conservation of mass, momentum, energy, and species concentrations: • Mass continuity: ∂V i = 0 (1) ∂xi where Vi = mean velocity component in xi-direction 3 xi = coordinate (for I=1, 2, 3, xi corresponds to three perpendicular axes). • Momentum: ∂ρV ∂ρVVij ∂p ∂ ⎡ ⎛ ∂V ∂Vj ⎞⎤ i +=−+⎢μ ⎜ i +⎟⎥ +−ρβ()TTg (2) t x xxeff ⎜ x x ⎟ oi ∂ ∂ jij∂ ∂ ⎣⎢ ⎝ ∂ j ∂ i ⎠⎦⎥ where ρ = air density Vj = velocity component in xj-direction p = pressure μeff = effective viscosity β = thermal expansion coefficient of air To = temperature in a reference point T = temperature g = gravity acceleration The last term on the right side of the equation is the buoyancy term. The turbulent influences are lumped into the effective viscosity is the sum of the turbulent viscosity, μt, and laminar viscosity, μ: μμeff=+ t μ (3) The Prandtl-Kolmogorov assumption, the turbulent viscosity expresses as the product of turbulence kinetic energy, k, and turbulent macroscale, l, that is a proper length scale for turbulence interactions: 12/ μρt = Cklν (4) where Cν = 0.5478, an empirical constant. Depending on how to solve the unknown parameters k and l, eddy-viscosity models have different forms. The simplest model is probably the Prandtl’s mixing-length model (Prandtl 1926) and complicated ones use multi-equations for turbulence transport. The standard k-ε model (Launder and Spalding 1974) is the most widely used two-equation model. In this paper, we use a single algebraic function to express the turbulent viscosity as a function of local mean velocity, V, and a length scale, l: μt = 0.03874 ρ V l (5) This equation has no adjustable constants between different flow conditions. 4 • Energy: To determine the temperature distribution and the buoyancy term in Equation (2), the conservation of energy must be solved. ∂ρT ∂ρVTj ∂ ⎛ ∂T ⎞ ⎜ ⎟ +=⎜ΓT,eff ⎟ + q/Cp (6) ∂t ∂xxjj∂ ⎝ ∂x j ⎠ where ΓT,eff = effective turbulent diffusion coefficient for T q = thermal source Cp = specific heat In our work we have estimated the effective diffusive coefficient for temperature in Equation (6), ΓT,eff, by: μeff ΓTeff, = (7) Preff where the effective Prandtl number, Preff, is 0.9. • Species concentrations: For determination of pollutant or water vapor concentration distribution the conservation of mass must be combined with the equation of transfer of the species. ∂ρC ∂ρVCj ∂ ⎛ ∂C ⎞ ⎜ ⎟ +=⎜ΓCeff, ⎟ + SC (8) ∂t ∂xxjj∂ ⎝ ∂x j ⎠ where C = species concentration ΓC,eff = effective turbulent diffusion coefficient for C SC = source term of C Similar method to the energy equation is used to determine the effective diffusive coefficient for species concentration in Equation (8), ΓC,eff: μ eff ΓCeff, = (9) Sceff where effective Schmidt number, Sceff, is 1.0. Equations (1) to (9) form the new simplified model. Boundary Conditions 5 Boundary conditions are necessary for the mathematical solution of the governing flow equations. There are three types of boundaries of practical importance: free boundary, symmetry surface, and conventional boundary. • Free boundary The boundary surface may be adjacent to an inviscid stream.
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